Science: Innate Sense of Numbers

Rob Stein
Washington Post Staff Writer
Monday, September 8, 2008; 11:00 AM

Washington Post staff writer Rob Stein and Justin Halberda, an associate professor of psychological and brain sciences at Johns Hopkins University were online on Monday, Sept. 8 at 11 a.m. ET to discuss how new research that shows a link between a primitive, intuitive sense of numbers and performance in math classes. The findings could lead to new ways to help children struggling in school.

Read today's Science Page story: How One's 'Number Sense' Helps With Mathematics.

A transcript follows.


Rob Stein: Thanks for joining us today to talk about "number sense" -- that intuitive ability we have to estimate numbers. Joining me today is Justin Halberda of Johns Hopkins University, who led the team that did this fascinating new research about how our number sense relates to our ability to learn math. I see there are already questions waiting. So we'll get right to it.


Sydney, Australia: In your opinion, how may a common phenomenon such as "mathematics anxiety" fit into the model of innate number sense in young children and their mathematics achievement? I am particularly interested in the initial years of schooling were formal mathematics instruction begins.

Rob Stein: This study didn't examine that issue directly. I'd imagine that "math anxiety" could clearly interfere with a child's ability to use math, and could potentially be caused or exacerbated by difficulties caused by their "number sense."


Rockville, Md.: In our multisensory math course for teachers and tutors at ASDEC, we advocate the use of manipulatives for all students not just those who struggle. Would you agree that this research supports using manipulatives and helping children see concrete representations of quantity to help them to develop a sense of numberness. We want them to understand, for example, the "fiveness" of five and how it is related to fifty and to five hundred and five hundred thousand. I know there are references to the Concrete-Representational-Abstract approach being promising in a recent national report and wwe are always looking for research based best practices. Marilyn Zecher, MSM Instructor

Rob Stein: One of the key questions about this research is the "chicken-and-egg" question. In other words, does the better number sense influence the ability to learn math? Or does a person's number sense improve from learning math? Prof. Halberda is already trying to answer that question.


Reston, Va.: Could you recommend a book that would help in understanding... how to read an equation?

I know people who can look at an equation and instantly understand, for example, why a particular variable is squared, or why a particular constant is present.

They can talk about the variables and their relationships using non-mathematical language to explain what the equation is modeling. I personally would love to get to that point of understanding.

Justin Halberda: At that type of level of complexity, a lot of what we are doing when we think mathematically is compressing and transforming information. Seeing a pattern, a shape, instead of all the individual variables that make up the shape is one type of compression. It is similar to chunking.


Seattle, Wash.: Logical thinking and problem-solving (e.g. breaking a problem up into parts) play an important part in math. Has there been an attempt to test for these abilities at an early age and correlate with math?

Justin Halberda: Another aspect of my work is looking at logical thinking: what are our early foundations for thinking logically? Work that I've been doing suggests that 2 year olds have access to reasoning via Process of Elimination (aka Disjunctive Syllogism, Modus Tollendo Ponens). What is interesting here is that noone has ever sat a 2 year old down and said, "Hi Jimmy, today we will learn Modus Tollendo Ponens. When you are unsure of an answer, you can infer that answer by a systematic elimination of all other possibilities." That 2 year olds will spontaneously engage in this kind of reasoning suggests that it is part of an early developing reasoning system that we all have access to.


Rockville, Md.: Please consider if playing "catch" might help with strengthening math skills. Volleyball and Frisbee really push the skill of estimating where an object will land. In volleyball, some players are very good at quickly calling a ball as landing "in" or "out". Frisbee players & some dogs also do this well with a quick look at the Frisbee. This essentially is a quick estimate of a difficult calculus problem.

Thanks, - Bill

Rob Stein: Interesting thought. My guess would be that these would be separate cognitive skills, with the ability to play volleyball or frisbee being influenced more by visual and spatial abilities. The researchers did look at how other skills, such as IQ and visual and spatial abilities, influenced number sense and math ability in school, and found that the relationship between number sense and math in school appears to be independent of those other factors.


Maryland: Really interesting and believable finding. I wonder about a different hypothesis. Take kids who all do well in math class. Do the ones with tiptop number sense use different strategies from the ones with average number sense?

I ask because of an introspective anecdote (yes, newspaper readers always react this way, sorry). I think I have really good number sense, and consistent with the trend in the study I was always top of my school in math all the way through high school calculus... but then when I tried some more formal math in college it wasn't a pretty sight! As an adult, in terms of real life quantitative skills I'm very competent, but I would never be another Karl Friedrich Gauss (to put it mildly).

Justin Halberda: I think this raises similar issues to the post from Reston, Va. And you are highlighting something that we didn't have space to talk about in the published paper. That is, the "number sense" should only correlate with and help certain aspects of mathematics (the ones that would likely rely on a sense of a number line). These include simple transformations like addition & subtraction and any cases where you are thinking about numbers. But for higher order mathematics many other factors come into play. Strategy use is one of them. Thinking logically is another.


Princeton, N.J.: Hi, I am a retired research mathematician. I think part of the problem is that more people are using faith based reasoning systems, I mean more than just regilious faith. It their argument has a gap, they take a "leap of faith." While getting people to think has always been a problem. I believe the tendacy to avoid research and thought is on the rise. This becomes very obvious when you try to make a mathematical argument to a person whose mind is made up. Bertrand Russell said it best. "Most people would rather die than think. And they do."

Rob Stein: Well, that's certainly gets at one of the hot-button political and social debates of our time -- with the debate over evolution being the obvious case in point.


Justin Halberda: Concerning Rob's answer to the earlier post about multisensory instruction, manipulative etc. The number sense is engaged by every day numerical discrimination. So, it will be activated by groups of objects for example. We do not yet know whether engaging number concepts via manipulatives will improve number sense directly, but it is certainly conceivable that it would.


Springfield, Va.: The article in the Post this morning had many interesting connections to what many may not think of as math, like getting on a crowded bus, but the two school districts I have worked for in the last 20 years have both stressed the foundation of number sense for their math programs so I am not clear about what the new information is that the study shows. Could you clarify this?

Rob Stein: The idea that we have this "number sense" has been known for a long time. And research has shown that animals have it too. What's been unclear is the relationship between this number sense and the ability to learn math in school. This research shows that there is an association.


Philadelphia, Pa.: How does dyslexia play into this? I ask because there are times when I zone out where I seem to be good at math, but some weird dyslexia kicks in. I remember thinking I had aced an Accounting exam, only to discover the teacher failed me because I had the columns reversed on the wrong side, even though the math and caluclations were correct. Can math skills cause a los of other types of skills when the brain is operating?

Justin Halberda: There is a disability associated specifically with mathematics called dyscalculia. This impacts around 5% of the population (as frequent as dyslexia), but much less is currently known about dyscalculia. There is some evidence that there are multiple (perhaps 3) different forms of dyscalculia. In one form a person would suffer from a very inaccurate number sense. We have some evidence for this in our own data with children with very low number sense (doing poorly on the blue and yellow dot task) doing very poorly in school mathematics. Another form of dyscalculia is similar to what you are describing, i.e., having difficulty with symbolic forms of mathematics, keeiping written numbers straight and orderly.


Spatial relations and numbers?: It's common to lump together a talent for numbers with a spatial sensibility. Does your research support this?

I'm a statistician but find most graphs much inferior to a verbal description -but maybe that says more about common graphic techniques.]

Rob Stein: The researchers did find that the relationship between number sense and math performance in school was independent of other factors, including visual/spatial ability.


Rockville, Md.: Will you also be researching whether or not numeracy can be developed later in life or if it is solely an innate ability? This could have huge implications for remedial programs and approaches for special education.

Justin Halberda: I currenlty have work going on with high-school students, college students and the very young kids. We're trying to figure out the impact of the number sense at each of these levels. It is an exciting moment with lots of work to do.


Rockville, Md.: There is also the problem of people waiting in line and trying to estimate if they should wait or leave.

Rob Stein: Yes, I had this problem myself while waiting for a crab cake sandwich at Eastern Market. I decided not to wait, and by the time the line shortened up they were out of crab cakes.


Rockville, Md.: I think one of the factors of this is how people pay large attention to small factors. You may have a one in a million chance of radiation from a counter top of rock, but many will replace it to avoid that risk. Or they can say we have two parts per billion in our water and people are concerned. As far as I am concerned "parts per billion" are like ghosts.

Rob Stein: That gets into a very interesting area -- risk assessment. A lot of research has shown that people are much more willing to accept risk that they have control over than risk that they don't have control over. That helps explain why people who smoke or drive without seatbelts nonetheless worry about risks that are far smaller than those posed by cigarettes and car accidents.


Rockville, Md.: I was intrigued by your use of color in the dot patterns of the research. I have heard of other research on numeracy also using dot patterns. I am wondering how the use of color and size might have impacted the participant's perception in your experiments and why you chose to use two (or more?)different colors and sizes. We speak of figure ground difficulties in the field of learning disabilities and I would think that added a layer of complexity. Also, Did you in any way make allowances for for any degree of color blindness.

Justin Halberda: Color is processed very early in visual processing and is one of the cues that we use well in the natural world for choosing groups of items from among a background. My lab recently published a paper about this kind of selection ( We used blue and yellow here to avoid color blindness issues (they differ in luminance as well as color). The kinds of figure ground segmentation you are bringing up would certainly be a concern. But they would also have been an issue in many of our control tasks. So if a child had difficulty in our task because of such issue, we would also have seen this in the control tasks. The different sizes were so that children could not rely on total amount of area on the screen instead of number. We want to make sure that people are using number to make their decision.


Rockville, Md.: Not a question, just a comment. I've always thought that people's ability to estimate numbers of objects to be quite poor. Take a very large crowd of people for example. We might be able to estimate the crowd size if they're in a stadium, but that probably reflects reasoning along the lines of "well, such-and-such stadium can hold 'x' many people, so this stadium, being smaller, probably has 'y' number of people." Take the same crowd and put them on a plain or line them along the banks of river and it becomes much, much harder to estimate numbers with any sort of accuracy.

Also, I remember a soil analysis class I once took in which we had to quantify certain visually identifiably heterogeneous mixtures. It was all but impossible to say "these visibly larger particles are 'x' percent of the whole" without pictures showing us what a 1% distribution looked like, a 10 percent distribution, a 25 percent distribution, etc.

Justin Halberda: In fact, estimation error increases linearly as a function of the number to be estimated. This comes from the way that the Approximate Number System (the cognitive system we studied) stores information about numbers. So, at the level of people in the stadium, this number sense would be very error prone (notice that those large numbers are not what we would typically be using a number sense for in our day to day lives). I think you are right, at these large numbers, strategy use and other forms of reasoning become key.


Takoma Park, Md.: Dear Justin, Nice comment about the 2-year-old and logic classes. When I was in kindergarten, I took a weird class in my school called "Logic." No joke. I learned about the contrapositive (I forget what they called it, but that's what it was) and I've never forgotten.

Rob Stein: Thanks for this! I'll pass it along...


Rockville, Md.: I have read that the human brain can recognize a quantity of up to four without counting but that over four, there needs to be a pattern. Did any of your past research or does any of your current research tie pattern recognition to numeracy?

Justin Halberda: You are right, but the research suggest that it is not pattern recognition. Rather, humans and other animals share an ability to pay attention to about 3-4 things at once. Our attention is constantly jumping around paying attention to 3-4 objects at any one time. It is this ability that allows us to rapidly and accurately say whether there is 1, 2, 3, or 4 objects in front of us (or on a screen of dots). above 4, the Approximate Number System is what we use to estimate. You can find lots of papers about these different number systems (and about how they work in babies) at our lab webpage (


Rob Stein: Thanks everyone for joining in our discussion today. And I wanted to thank Prof. Halberta to helping out. This is obviously fascinating research and we look forward to hear more about what his future work reveals.

Justin Halberda: I very much enjoyed it. Thanks everyone for the interest!


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